ScalingStacks

Lemma 6.2.9. Given $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$, there is a bijection

$$\tilde{D}(\sigma)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\{\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)\ |\ \sigma^{\prime}<\sigma,\ \ell(\sigma^{\prime})=\ell(\sigma)-1\},\ (i_{1},i_{2})\mapsto\sigma^{i_{1},i_{2}}.$$

Note that

$$\{\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)\ |\ \sigma^{\prime}<\sigma,\ \ell(\sigma^{\prime})=\ell(\sigma)-1\}=\{\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)\ |\ \sigma^{\prime}<\sigma,\ \deg(\sigma^{\prime})=\deg(\sigma)+1\}.$$

Given $(i_{1},i_{2})\in L(\sigma)$, we have $(i_{1},i_{2})\in D(\sigma)$ if and only if $\deg_{D}(\sigma)=\deg_{D}(\sigma^{i_{1},i_{2}})-1$ for some subset (equivalently, for any subset) $D$ of $\{1,\ldots,n\}\times\{\pm 1\}$ that embeds in its projection on $\{1,\ldots,n\}$.